GEO 



Tn tills feiife we fay, geometrical foliition in coiitra- 

 ctifliiidlion to a mechanical, or inftrumental fi)!ution, 

 ivhore the problem is only folvcd by ruler and com- 

 paffcs. 



The fame term is likcwife ufed in oppofition to all indireft 

 and inadequate kinds of folutions, as by infinite fcriefes, 

 &c. 



We have no geometrical way of finding the quadrature of 

 the circle, the dupHcature of the cube, or two mean pro- 

 portionals ; but mechanical ways, and others, by infinite fc- 

 riefes, we have. 



The ancients, Pappus informs us, in vain endeavoured at 

 the trifeAion of an angle, and the finding out of two mean 

 proportionals by a right line, and a circle. Afterwards they 

 began to confider the properties of feveral other lines ; as 

 the conchoid, the ciffoid, and the conic fetlions ; and by 

 fome of thefe endeavoured to folve thofe problems. At 

 kngth, having more thoroughly examined the matter, 

 and the conic feftions being received into geometry, 

 they dillinguiflied geometrical problems into three kinds ; 

 viz. 



I. PItvie ones, which, deriving their original from lines on 

 a plane, may be regularly folved by a right line, and a 

 circle. 



z. SoFiJ ones., which are folved by lines deriving their 

 original from the confideratiofi of a folid ; that is, of a 

 cone. 



3. L'lnear ones, to the folution of which are required lines 

 more compounded. 



Accordingto this diftinftion we are not to folve folid pro- 

 blem.s by other lines than the conic feftions ; efpecially if no 

 other lines but right ones, a circle, and the conic feclions, 

 jnuft be received into geometry. 



But the moderns, advancing much farther, have received 

 into geometry all lines that can be exprefled by equations ; 

 and have di'Hnguilhed, according to the dimenfions of the 

 equations, thofe lines into knids ; and have made it 

 a law, not to conftruft a problem by a line of fupe- 

 rior kind, that may be conllrufted by one of an inferior 

 kind. 



Geometrical Square. See Square. 

 Gf.OMETRiCAL Tci/z/t". See Plain Talk. 

 GEOMETRICALLY PROPOKTioxAL.s,are quantities 

 in continual proportion ; or which proceed in the fame con- 

 ftantratio: as 6, 12, 24, 48, 96, 192, &c. 



They are thus called, in contradiftinclion to cqui-different 

 quantities ; whicli are called, though fomewhat improperly, 

 ar'ithnieticaHy proportionals. 



GEOMETRY, the fcience, or doClrine of extenfion, 

 or extended things ; that is, of lines, furfaces, or fo- 

 lids. 



The word is Greek y^iy-TTpi-^, formed of ■>rz or •>>■, earth, 

 and u!T|);i', meafure ; it being the neceility of meafuring the 

 earth, and the parts and places thereof, that gave the firfl 

 occafion to the invention of the principles and rules of this 

 art ; which has fincc been extended and applied to numerous 

 other things ; ir.fomuch that geometry, with arithmetic, is 

 now tlic general foundation of all mathematics. 



Herodotus, lib. ii. p. 102. edit. WelTelingii, Diodorus, 

 lib. i. ^81, or vol. i. p. 91. edit. Amft. 1746. and Strabo, 

 lib. xvii. vol. ii. p. 1139. edit. Amft. 1707. affert, that the 

 Eg\'pt:ans were the firfl inventors of geometry ; and 

 that the annual inundations of the Nile were the occafion of 

 it ; for tliat river bearing away all the bounds and landmarks 

 of men's eftates, and covering the whole face of the country, 

 the people, fay they, were obliged to diftinguilh their lands 

 by the confidcratioiiof their figure and quantity ; and thus, 



GEO 



by experience and habit, formed themfelvcs a metacd, or 

 art, which was the origin of geometry. A farther con- 

 templation of the drauglits of figures, of fields thus laid 

 down, and plotted in proportion, might naturally enough 

 lead them to the difcovery of fome of their excellent and 

 wonderful properties ; which fpeculution contmually im- 

 proviiio-, the art became gradually improved, as it continues 

 to do to this day. Jofephus, however, feem.s to attribute 

 the invention to the Hebrews : and others, among the an- 

 cients, make Mercury the inventor. Polyd. Virgil, De In- 

 vent. Rer. lib. i. cap. i?. 



From Egypt geometry pafled into Greece, being carried 

 thither, as fome fay, by Thalcs ; where it was much cul- 

 tivated and improved by himfilf, Pythagoras, Anaxagoras 

 ofClazomene, Hippocrates of Chios, and Plato, who tefti- 

 ficd his conviftion of the ncccfruy and importance of geome- 

 try in order to the fuccefsful lludy ot philolophy by the 

 following infcription on the door of his academy, tin: 

 a' xfj.:-7j tTTc; si7-=iTi', let no one ignorant of geometry enter here. 

 Plato, conceiving that geometry was too mean and relf ridted 

 an appellation for this fcience, fubftituted for it the more 

 exteniive name of " Menfuration ;'' and others have denomi- 

 nated it " Pantometry.'' Other more general and compre- 

 henfive appellations are more fuitable to its extent, more 

 efpeciallv in the prefcnt advanced ftate of the fcience ; and 

 accordinglv fome have defined it as the Science of inquiring, 

 inventing, and demonllrating all the affeftions of magnitude. 

 Proclus calls it the knowledge of magnitudes and figures, 

 with their limitations ; as alio of their ratios, affections, 

 pofitions, and motions of every kind. About fifty years 

 after Plato, lived Euclid, who coUefled together all thofe 

 theorems which had been invented by his predecefibrs in 

 Egypt and Greece, and digeiled them into fifteen books, 

 CHtitled the Elements of Geometry ; and thofe propofitions 

 which were not fatisfaftorily proved, he more accurateiv 

 demonftrated. (See Euclid.) The next to Euclid of 

 thofe ancient writers, whofe works are extant, is ApoFio- 

 nius Pergieus, who flouriflied in the time of Ptolemy Eue r- 

 getes, about two hundred and thirty years before Chrili:, and 

 about a hundred years after Euclid. (See his biographical 

 article. ) The third ancient geometer, whofe writings re- 

 main, is Archimedes of Syracufc, who was famous about 

 the fame time with Apollonius. (See AiiciiiMEnns.) V.'e 

 can only mention Eudoxus of Cnidus, Arcliytas of Taren- 

 tum, Philolaus, Eratofthenes, Arillarchus of Samos, Dino- 

 ftratus, the inventor of the quadratrix, Menechmus, his 

 brother and the difciple of Plato, the two Arilleufes, Conon, 

 Thrafideus, Nicoteles, Leon, Theudius, Hermotimus, and 

 Nicomedes, the inventor of the conchoid ; befides whom, 

 there are many other ancient geometers, to whom this fcience 

 is indebted. 



The Greeks continued their attention to geometry even 

 after they were fubdued by the Romans. Whereas tlie 

 Romans themfelves were fo little acquainted with this 

 fcience, even in tlie niofl flourifhing time of their republic, 

 that they gave the name of mathtmaticians, as Tacitus in- 

 forms us, to thofe who purfued the chimeras of divination 

 and judiciary aftrology. Ner v.ere they more difpofed to 

 cultivate geometry-, as we may reafonably imagine, during 

 the decline, and after the fall of the Roman empire. The 

 cafe was different \vith the Greeks ; among whom we find 

 many excellent geometers finee the commencement of the 

 Chrillian era, and after the tranflation of the Roman 

 empire. Ptolemy lived under Marcus Aurelius ;' and we 

 have extant the works of Pappus of Alexandria, who lived in 

 the time of Theodofius ; t'le commentary of Eutocius, tiie 

 Afcalonite, who lived about th<? year of Chriil 540, on 

 3 Archimedes's 



